Unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1.[1] Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. In topology, it is often denoted as S^{1} because it is a onedimensional unit nsphere.[2][note 1]
Trigonometry 


Reference 

Laws and theorems 
Calculus 

If (x, y) is a point on the unit circle's circumference, then x and y are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, x and y satisfy the equation
Since x^{2} = (−x)^{2} for all x, and since the reflection of any point on the unit circle about the x or yaxis is also on the unit circle, the above equation holds for all points (x, y) on the unit circle, not only those in the first quadrant.
The interior of the unit circle is called the open unit disk, while the interior of the unit circle combined with the unit circle itself is called the closed unit disk.
One may also use other notions of "distance" to define other "unit circles", such as the Riemannian circle; see the article on mathematical norms for additional examples.
In the complex plane
The unit circle can be considered as the unit complex numbers, i.e., the set of complex numbers z of the form
for all t (see also: cis). This relation represents Euler's formula. In quantum mechanics, this is referred to as phase factor.
Trigonometric functions on the unit circle
The trigonometric functions cosine and sine of angle θ may be defined on the unit circle as follows: If (x, y) is a point on the unit circle, and if the ray from the origin (0, 0) to (x, y) makes an angle θ from the positive xaxis, (where counterclockwise turning is positive), then
The equation x^{2} + y^{2} = 1 gives the relation
The unit circle also demonstrates that sine and cosine are periodic functions, with the identities
for any integer k.
Triangles constructed on the unit circle can also be used to illustrate the periodicity of the trigonometric functions. First, construct a radius OA from the origin to a point P(x_{1},y_{1}) on the unit circle such that an angle t with 0 < t < π/2 is formed with the positive arm of the xaxis. Now consider a point Q(x_{1},0) and line segments PQ ⊥ OQ. The result is a right triangle △OPQ with ∠QOP = t. Because PQ has length y_{1}, OQ length x_{1}, and OA length 1, sin(t) = y_{1} and cos(t) = x_{1}. Having established these equivalences, take another radius OR from the origin to a point R(−x_{1},y_{1}) on the circle such that the same angle t is formed with the negative arm of the xaxis. Now consider a point S(−x_{1},0) and line segments RS ⊥ OS. The result is a right triangle △ORS with ∠SOR = t. It can hence be seen that, because ∠ROQ = π − t, R is at (cos(π − t),sin(π − t)) in the same way that P is at (cos(t),sin(t)). The conclusion is that, since (−x_{1},y_{1}) is the same as (cos(π − t),sin(π − t)) and (x_{1},y_{1}) is the same as (cos(t),sin(t)), it is true that sin(t) = sin(π − t) and −cos(t) = cos(π − t). It may be inferred in a similar manner that tan(π − t) = −tan(t), since tan(t) = y_{1}/x_{1} and tan(π − t) = y_{1}/−x_{1}. A simple demonstration of the above can be seen in the equality sin(π/4) = sin(3π/4) = 1/√2.
When working with right triangles, sine, cosine, and other trigonometric functions only make sense for angle measures more than zero and less than π/2. However, when defined with the unit circle, these functions produce meaningful values for any realvalued angle measure – even those greater than 2π. In fact, all six standard trigonometric functions – sine, cosine, tangent, cotangent, secant, and cosecant, as well as archaic functions like versine and exsecant – can be defined geometrically in terms of a unit circle, as shown at right.
Using the unit circle, the values of any trigonometric function for many angles other than those labeled can be calculated without the use of a calculator by using the angle sum and difference formulas.
Circle group
Complex numbers can be identified with points in the Euclidean plane, namely the number a + bi is identified with the point (a, b). Under this identification, the unit circle is a group under multiplication, called the circle group; it is usually denoted On the plane, multiplication by cos θ + i sin θ gives a counterclockwise rotation by θ. This group has important applications in mathematics and science.
Complex dynamics
The Julia set of discrete nonlinear dynamical system with evolution function:
is a unit circle. It is a simplest case so it is widely used in the study of dynamical systems.
Notes
 Confusingly, in geometry a unit circle is often considered to be a 2sphere—not a 1sphere. The unit circle is "embedded" in a 2dimensional plane that contains both height and width—hence why it is called a 2sphere in geometry. However, the surface of the circle itself is onedimensional, which is why topologists classify it as a 1sphere. For further discussion, see the technical distinction between a circle and a disk.[2]
References
 Weisstein, Eric W. "Unit Circle". mathworld.wolfram.com. Retrieved 20200505.
 Weisstein, Eric W. "Hypersphere". mathworld.wolfram.com. Retrieved 20200506.
See also
 Angle measure
 Pythagorean trigonometric identity
 Riemannian circle
 Unit angle
 Unit disk
 Unit sphere
 Unit hyperbola
 Unit square
 Turn (unit)
 ztransform